Special Session 137: Fractional Calculus and Advanced Applications in Complex and Nonlinear Systems

Fractional-Order Operational Matrix Method for Eigenvalue Analysis of Nonsingular Second-Order Sturm-Liouville Problems

Lana Abdelhaq United Arab Emirates University United Arab Emirates Co-Author(s):    Lana Abdelhaq, Sondos M. Syam, Muhammad I. Syam

Abstract: This article focuses on the theoretical and numerical analysis of the eigenvalues associated with a non-singular fractional second-order Sturm-Liouville problem. We employ a fractional-order modified operational matrix method to approximate the eigenvalues, transforming the Sturm-Liouville problem into a sparse, non-singular linear system, which is then solved using the continuation method. Theoretical findings for the problem are presented and proven, with the fractional derivative defined in the Modified Atangana-Baleanu Fractional Derivative sense. The numerical results validate the accuracy of the proposed algorithm.

Eigenvalue Problems with ABC Fractional Derivatives: A Numerical Approach Using Operational Matrix Methods

Mohammed M A Abuomar UAE Universities Jordan Co-Author(s):    Mohammed Abu Omar, Sondos M. Syam, Muhammed Syam

Abstract: In this talk, we will investigate the eigenvalue problem involving ABC fractional derivatives. The numerical approach will be based on the operational matrix method. We will explore the main properties of the eigenvalues and the corresponding eigenfunctions. Both theoretical and numerical results will be presented.

Fractional logistic equation with variable kernel in the Caputo sense

Madhukant Sharma Dhirubhai Ambani Institute of Information and Communication Technology Gandhinagar, Gujarat India Co-Author(s):    Madhukant Sharma, Sharad Dwivedi, Sanjeev Singh

Abstract: We consider a Caputo-type fractional derivative of order $q\in (0,1]$ with a variable kernel $\psi$, which has been introduced in the literature for its efficacy in analyzing real-world models through appropriate selection of fractional derivatives. This inspired us to incorporate this generalized fractional operator in the logistic differential equation, pivotal in studying population dynamics. We identify the equilibrium points and evaluate their stability using the $\psi-$Laplace transform technique. The proof of the solution`s existence and uniqueness is achieved through employing the fixed-point theorem. Additionally, we derive the representation for the analytic solution as an infinite series by introducing the fractional $\psi-$series expansion, which has a positive radius of convergence. To conclude, by considering various kernels, we demonstrate the utility of the truncated series in closely approximating the analytical solution for different values of $q$.

Fractional logistic equation with variable kernel in the Caputo sense

Madhukant Sharma Dhirubhai Ambani Institute of Information and Communication Technology Gandhinagar, Gujarat India Co-Author(s):    Madhukant Sharma, Sharad Dwivedi, Sanjeev Singh

Abstract: We consider a Caputo-type fractional derivative of order $q\in (0,1]$ with a variable kernel $\psi$, which has been introduced in the literature for its efficacy in analyzing real-world models through appropriate selection of fractional derivatives. This inspired us to incorporate this generalized fractional operator in the logistic differential equation, pivotal in studying population dynamics. We identify the equilibrium points and evaluate their stability using the $\psi-$Laplace transform technique. The proof of the solution`s existence and uniqueness is achieved through employing the fixed-point theorem. Additionally, we derive the representation for the analytic solution as an infinite series by introducing the fractional $\psi-$series expansion, which has a positive radius of convergence. To conclude, by considering various kernels, we demonstrate the utility of the truncated series in closely approximating the analytical solution for different values of $q$.

An accurate method for solving time-fractional nonlinear PDEs with proportional delays

Muhammed I Syam UAE University United Arab Emirates Co-Author(s):    Sondos M. Syam

Abstract: The operational matrix method is an effective technique for addressing fractional initial or boundary value problems. In this paper, we introduce a modified version of the operational matrix method that eliminates the necessity of solving a large system to determine the coefficients of the solution. Instead, we obtain the coefficients explicitly and iteratively as functions of previously calculated coefficients. We develop this new approach and prove that the iterative process produces a sequence of functions that converges uniformly to the unique solution of the given system. Furthermore, we demonstrate the existence and uniqueness of the solution. To validate the proposed method, we apply it to several numerical examples and investigate various applications. We evaluate the accuracy of the solution using error measures such as the $L_2$-error and minimization error. A comparison with existing methods shows that the modified approach is not only more accurate but also computationally efficient, easier to implement, and requires less computational time than the conventional operational matrix method.

Fractional Riccati Systems: A Numerical Approach with Error Analysis

Sondos M Syam Universiti Malaya Malaysia Co-Author(s):    Zailan Siri and R. Md. Kasmani

Abstract: In this study, we explore the solution of a fractional system of Riccati equations, which plays a significant role in various scientific applications, including control theory. To address this system, we employ the operational matrix method. Initially, the block-pulse operational matrices aid in transforming the nonlinear fractional-order Riccati-differential problem into a system of algebraic equations. One of the key advantages of this approach is that it offers a cost-effective framework for setting up the equations without relying on projection methods such as Galerkin, collocation, or similar techniques. Furthermore, we establish the convergence of the approximate solution obtained through the operational matrix method toward the exact solution. To demonstrate the efficacy of the proposed numerical method, we present two illustrative examples. The results show that the error is on the order of $\mathcal{O}(10^{-13})$. Additionally, the approximate solutions are shown to converge to the exact solutions for various values of $\gamma$. As $\gamma$ approaches one, the approximate solutions increasingly align with the solution for $\gamma = 1$.